2 Trigonometry—An Introduction Trigonometry is the study of the relationship between the angles and the sides of triangles.Example: Given a right triangle with legs 3 and 4, what is the hypotenuse?Solution: Since in a right triangle, (leg 1)2 + (leg 2)2 = (hypotenuse)2 i.e., a2 + b2 = c2. We have= x2 = x2 25 = x2 x = 5.What does trigonometry have to do with this? Nothing and everything!It’s true that we don’t need trigonometry to find the hypotenuse when we are given two legs of a right triangle. However, we need trigonometry when we are given the following:Given a right triangle with legs 3 and 4, what is the measure of the angle opposite of the leg with length 4?leghypotenuse34x34x
3 SOH CAH TOA Trigonometry—Sine, Cosine and Tangent When we want to solve a side and/or an angle of a right triangle, we tend to use English alphabets for the sides (a, b, c, x, etc.) and Greek alphabets for the angles. Common Greek letters used to denote angles include (theta), (alpha), (beta) and (gamma).To solve an angle, we must use one of the following three definitions in trigonometry.Definitions:In trigonometry, we have 3 basic definitions—sine, cosine and tangent.We define the sine of an acute angle of a right triangle to be the ratio of the opposite leg to the hypotenuse.We define the cosine of an acute angle of a right triangle to be the ratio of the adjacent leg to the hypotenuse.We define the tangent of an acute angle of a right triangle to be the ratio of the opposite leg to the adjacent leg .To remember these three ratios, we haveSOH CAH TOA34xAdjacent legOpposite legHypotenuse
4 Trigonometry—Applying Sine, Cosine and Tangent Of course, in application, we have to use the appropriate one to solve the angle. For example, to solve angle of the triangle on the right, we need to use tangent, since we only know the opposite side of angle , which is 4, and the adjacent side of angle , which is 3. This is what we do:(Set up the equation using the tangent definition)(To cancel tan, we use its inverse, which is tan–1) (1)(Result obtained by calculator: 2nd Tan (4 ÷ 3) = )Notice that we actually do know the remaining side (i.e., the hypotenuse) is 5. So, in this case, it doesn’t matter which definition (sine, cosine, or tangent) we use. We can use sine and cosine to find angle if we want to.If we use sine: If we use cosine:43Tan–1Note:1. Tangent (tan) is not a number, so we don’t and we can’t divide both sides by tan. Tangent is a function, to undo it, we use inverse tangent, which is commonly denoted by tan–1 or arc tan. Similarly, inverse sine is denoted by sin–1, and inverse cosine, by cos–1.
5 Trigonometry—Solving a Right Triangle We say a triangle has 6 parts—3 sides and 3 angles. In almost every case, only 3 parts are needed to be given, and we can solve the other 3 parts based on the 3 given parts.Examples:135yy7x32°817x640°xr
6 Trigonometry—Special Right Triangles There are two kinds of special right triangles. They are special because, unlike previous examples, we don’t need to use a calculator.Examples:2y60°r645°xr5rx30°yr345°
7 Trigonometry—Let’s Make a Table 0° 30° 45° 60° 90° sin cos tan Degenerate Right Triangle: 0°-90°-90°0°30°-60°-90° Right Triangle:30°60°45°-45°-90° Right Triangle:45°45°90°
8 Trigonometry—Word Problems 1. The Chrysler Building, once the tallest building in the world, has a spire on top of its roof. A surveyor tries to determine the height of the spire by taking two observations on the ground that is 412 ft from the center of the building. The angle of elevation to the roof is 66.0 and angle of elevation to the tip of the spire is 68.5. What is the height of the spire?2. A 13-ft. ladder is leaning against of a wall. If the tip of the ladder is 12 ft. above the ground, what is the angle formed by the ground and the bottom of the ladder? If the bottom of the ladder is pushed 1 ft. toward the wall, will the angle (formed by the ground and the bottom of the ladder) increase or decrease? By how many degrees?
9 Trigonometry—Angles in Standard Position Initially, trigonometry is devoted to the study of the relationship between the angles and the sides of right triangles.It then expands to include angles and sides of ANY triangles.It also expands to include angles that can’t be angles of a triangle. For example, angles > 180° or < 0°.sidevertexAngles in Standard PositionAn angle is a union of two rays with a common endpoint— the rays are called sides and the endpoint is called a vertex.An angle in standard position is an angle on the x-y plane with the vertex is the origin and a side is on the positive x-axis, which we called the initial side. The other side is called the terminal side.If the direction of the span of an angle (from the initial side to the terminal side) is counter- clockwise, the angle sustains a positive mea- sure. If the direction is clockwise, the angle sustains a negative measure.xyInitial SideTerminal Sidexy50°xy–50°
10 Trigonometry—Angles in Standard Position (cont’d) The initial side of an angle in standard position is always the positive x-axis.yyIf the terminal side of an angle lies in the first quadrant, it’s called a first quadrant angle or a quadrant one angle; if the terminal side is in the second quadrant, it’s a second quadrant angle or a quadrant two angle angle, and so on.Q I angleQ II anglexxyyIf the terminal side of an angle lies on either the x-axis or the y-axis (i.e., in between two quadrants, it’s called a quadrantal angle. The measures of these angles are 0°, 90°, 180°, 270°, 360°, etc. That is, they are _________________.xxQ III angleQ IV anglexy90°xy180°xy270°xy360°
11 Trigonometry—Coterminal Angles The initial side of any angle is always the positive x-axis. However, if two (or more) angles share the same terminal side, they are called coterminal angles.Find the indicated coterminal angle for the following angles:yxy127°yy216°304°41°xxx___°___°___°___°41° – ___ ° = ____° 127° – ___° = ____° ___ ° – 216° = ____° 304° – ___° = ____°What about coterminal angles?Coterminal angles differ by _____ or by ___________________.trig = trig coterminal (regardless of what quadrant is in)That is, trig = trig ( + 360°k) where k is any integer
12 Trigonometry—Reference Angles How to Find the Trigonometric Function Values of Angles in Standard Positionxy41°If the angle is in the first quadrant, we can find the trigonometric function value of that angle by dropping a perpendicular from the terminal side to the x-axis.If the angle is in the second (or third or fourth) quadrant, we can do the same thing, i.e., drop a perpendicular from the terminal side to the x-axis. Of course, the angle will be outside of the right triangle formed. However, we can use the angle that is inside the triangle, which is called the reference angle. Also, the lengths of the legs of the right triangle formed should be negative if they are indeed negative (the hypotenuse, however, is always positive).xy127°xy216°xy304°
13 Trigonometry—Reference Angles (cont’d) What about reference angles?Reference angles are always _________________.If the angle, , is in Q I, its reference angle is _____. If is in Q II, its reference angle is ______. If is in Q III, its reference angle is ______. If is in Q IV, its reference angle is ______.trig = trig ref up to the signThat is, either trig = trig ref or trig = –trig ref (depends on the quadrant is in) (1)refQ IQ IIQ IIIQ IVNote:1. The symbol means “is in.”If we know ref, how do we find if Q I: _______ Q II: _______ Q III: _______ Q IV: _______ref56°108°247°349°ref47° Q I39° Q II85° Q III61° Q IV
14 sin = cos = tan = sin = cos = tan = sin = cos = Trigonometry—ASTC (All Students Take Calculus)Let’s find the trigonometric function values of the following:3. 4.xy512ysin =cos =tan =sin =cos =tan =(–3, 4)xyysin =cos =tan =sin =cos =tan =xx(–2, –2)(4,–5)